Iris registration method for ophthalmic laser surgical procedures

ABSTRACT

In a laser cataract procedure that also corrects for astigmatism, an iris registration method compares an iris image of a patient&#39;s eye taken when the eye is not docked to a patient interface device with an iris image of the same eye that is docked to the patient interface, to calculate a rotation angle between the two images. The astigmatism axis of the eye is measured when the eye is not docked, and the measured axis is rotated by the calculated rotation angle to obtain a rotated astigmatism axis relative to the iris image of the docked eye. The laser cataract procedure is performed based on the rotated astigmatism axis. The rotation angle is calculated by optimizing a transformation that transforms the undocked iris image to match the docked iris image, where the transformation includes a dilation factor that accounts for different pupil dilation of the two iris images.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 62/755,122, filed on Nov. 2, 2018, which is incorporated herein byreference in its entirety.

BACKGROUND OF THE INVENTION Field of the Invention

This invention relates to an iris registration method for an ophthalmiclaser surgical procedure, and in particular, it relates to an irisregistration method for registering corneal astigmatism measurementresults to docked eye images.

Description of Related Art

Ultrashort (e.g. femtosecond) pulsed laser systems are used to performlaser cataract procedures, which includes using the laser beam to makeincisions on the surface of the eye such as the cornea or sclera, makeincisions on the lens capsule, and fragment the lens for easy removal.After the lens is removed, an intraocular lens (IOL) is implanted in thelens capsule. The same laser system may be used to correct cornealastigmatism while performing the cataract procedure, for example, bymaking arcuate relaxation incisions in the cornea or sclera to changethe tension in the cornea, and/or by using a toric IOL and accuratelyaligning the IOL relative to the axis of corneal astigmatism, providedthat the patient's corneal astigmatism (the a-sphericity, including theorientation of the steep meridian) is known. A patient's cornealastigmatism may be measured beforehand using the imaging capabilities ofthe laser system used to perform the cataract procedure, or on adiagnostic device that is separate from the laser system. When measuringthe astigmatism, the patient's eye is free of any mechanical contactwith the laser system or the diagnostic device, i.e., the eye is notmechanically coupled to the patient interface (PI) device that istypically used during laser cataract procedures. This is because thecornea is easily deformable and its sphericity will change temporarilywhen the eye is coupled to the PI device, preventing an accurateastigmatism measurement.

After the astigmatism is measured, the patient's eye is mechanicallycoupled to the patient interface device of the laser system (referred toas docking of the eye) in order to carry out the cataract procedure.Thus, the actual orientation of corneal astigmatism of the docked eye(the eye that is coupled to the PI) may be different from the previouslymeasured orientation because of possible cyclorotation anddocking-induced rotation of the eye.

Conventional means of registering the patient's axis of astigmatism(e.g. the steep meridian of the cornea) to the coordinate frame of thelaser system include visually evaluating the eye using a video image ofthe eye taken by an onboard imaging system and manually placing inkmarks on the eye. In another conventional method, the physician manuallyaligns fiducial features of the patient interface device to thepatient's eye. Sometimes the possible rotations of the eye are simplyignored, and the axis of astigmatism is aligned to the laser system'scoordinate frame without compensation for cyclorotation anddocking-induced rotation of the eye.

SUMMARY

Accordingly, the present invention is directed to an iris registrationmethod and related apparatus that substantially obviates one or more ofthe problems due to limitations and disadvantages of the related art.

An object of the present invention is to accurately determine theorientation of astigmatism of an eye that is docked to the laser system.

Additional features and advantages of the invention will be set forth inthe descriptions that follow and in part will be apparent from thedescription, or may be learned by practice of the invention. Theobjectives and other advantages of the invention will be realized andattained by the structure particularly pointed out in the writtendescription and claims thereof as well as the appended drawings.

To achieve the above objects, the present invention provides a laserophthalmic surgery system for treating a patient's eye, which includes:a laser source configured to generate a pulsed laser beam; an opticaldelivery system coupled to the laser source, and configured to receiveand direct the pulsed laser beam; a camera coupled to the opticaldelivery system and configured to obtain images of the eye; and animaging system configured to measure structures of anatomical componentsof the eye; a processor coupled to the laser source, the opticaldelivery system, the camera and the imaging system, the processorcomprising a non-transitory computer readable medium storing computerexecutable instructions configured to instruct the processor to performa process which includes: obtaining an undocked iris image of the eye,the undocked iris image having been taken when the eye is notmechanically coupled to any patient interface device; obtaining ameasured astigmatism axis orientation of the eye, the measuredastigmatism axis orientation having been measured when the eye is notmechanically coupled to any patient interface device; controlling thecamera to take a docked iris image of the eye when the eye ismechanically coupled to a patient interface device; computing a rotationangle of the eye between the docked and undocked iris images, including:defining a parameterized transformation which maps pixel positions in afirst iris image, which is one of the undocked and docked iris images,to corresponding mapped pixel positions in a second iris image, which isthe other one of the undocked and docked iris images, wherein theparameterized transformation includes a translation mapping usingtranslation parameters, a dilation mapping using a pupil dilationparameter, and a rotation mapping using a rotation angle parameter,wherein the dilation mapping maps a distance between a pixel and a pupilcenter to a mapped distance based on the dilation parameter whilemapping a limbus radius of a limbus of the eye to the limbus radiusitself; and optimizing the transformation by minimizing an error termthat represents a difference between pixel values of the first irisimage at its pixel positions and pixel values of the second iris imageat the corresponding mapped pixel positions, to obtain optimized valuesof the transformation parameters including an optimized rotation angle;and computing a rotated astigmatism axis orientation by rotating themeasured astigmatism axis orientation by the optimized rotation angle ina predetermined direction; and while the eye is mechanically coupled tothe patient interface device, controlling the laser source and theoptical delivery system based on the rotated astigmatism axisorientation to deliver the pulsed laser beam into the eye to correctastigmatism of the eye.

In another aspect, the present invention provides a method for treatinga patient's eye, implemented in a laser ophthalmic surgery system, themethod including: obtaining an undocked iris image of the eye, theundocked iris image having been taken when the eye is not mechanicallycoupled to any patient interface device; obtaining a measuredastigmatism axis orientation of the eye, the measured astigmatism axisorientation having been measured when the eye is not mechanicallycoupled to any patient interface device; controlling a camera of thelaser ophthalmic surgery system to take a docked iris image of the eyewhen the eye is mechanically coupled to a patient interface device;computing a rotation angle of the eye between the docked and undockediris images, including: defining a parameterized transformation whichmaps pixel positions in a first iris image, which is one of the undockedand docked iris images, to corresponding mapped pixel positions in asecond iris image, which is the other one of the undocked and dockediris images, wherein the parameterized transformation includes atranslation mapping using translation parameters, a dilation mappingusing a pupil dilation parameter, and a rotation mapping using arotation angle parameter, wherein the dilation mapping maps a distancebetween a pixel and a pupil center to a mapped distance based on thedilation parameter while mapping a limbus radius of a limbus of the eyeto the limbus radius itself; and optimizing the transformation byminimizing an error term that represents a difference between pixelvalues of the first iris image at its pixel positions and pixel valuesof the second iris image at the corresponding mapped pixel positions, toobtain optimized values of the transformation parameters including anoptimized rotation angle; and computing a rotated astigmatism axisorientation by rotating the measured astigmatism axis orientation by theoptimized rotation angle in a predetermined direction; and while the eyeis mechanically coupled to the patient interface device, controlling alaser source and a optical delivery system of the laser ophthalmicsurgery system based on the rotated astigmatism axis orientation todeliver the pulsed laser beam into the eye to correct astigmatism of theeye.

Preferably, the transformation is defined as:

$\begin{Bmatrix}x_{d} \\y_{d}\end{Bmatrix} = {{\overset{\rightarrow}{T}\left( \begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} \right)} = {{\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)*\begin{bmatrix}{\cos(\theta)} & {\sin(\theta)} \\{{- \sin}(\theta)} & {\cos(\theta)}\end{bmatrix}\ \left\{ {\begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} - \ \begin{Bmatrix}x_{{center} - u} \\y_{{center} - u}\end{Bmatrix}} \right\}} + \begin{Bmatrix}x_{{center} - d} \\y_{{center} - d}\end{Bmatrix}}}$

where x_(u) and y_(u) are x and y coordinates of a pixel position in thefirst one of the undocked and docked iris images, x_(d) and y_(d) are xand y coordinates of a corresponding transformed pixel position, e isthe pupil dilation parameter and

$\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)$

is the dilation mapping, θ is the rotation angle parameter, x_(center-u)and y_(center-u) are the translation parameters representing x and ycoordinates of a pupil center in the first one of the undocked anddocked iris images, x_(center-d) and y_(center-d) are x and ycoordinates of the pupil center in the second one of the undocked anddocked iris images, L is the limbus radius, and

R=√{square root over ((x _(u) −x _(center-u))²+(y _(u) −y_(center-u))²)}

is a radial position of the pixel in the first one of the undocked anddocked iris images with respect to the pupil center.

Preferably, the step of optimizing the transformation is performed usinga set of initial parameter values and a gradient Newton-Raphsoniterative method.

In another aspect, the present invention provides a computer programproduct comprising a computer usable non-transitory medium (e.g. memoryor storage device) having a computer readable program code embeddedtherein for controlling a data processing apparatus, the computerreadable program code configured to cause the data processing apparatusto execute the above method.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and areintended to provide further explanation of the invention as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically illustrates an ophthalmic laser surgery systemwhich may be used to implement embodiments of the present invention.

FIG. 2 schematically illustrates an ophthalmic laser surgery systemwhich may be used to implement embodiments of the present invention.

FIG. 3 schematically illustrates an overall procedure of the ophthalmicsurgical procedure according to an embodiment of the present invention.

FIG. 4 schematically illustrates an iris registration method accordingto an embodiment of the present invention.

FIGS. 5A-5H are examples that illustrate the effect of various factorsin the transformation according to an embodiment of the presentinvention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

A laser surgery system 10 that may be used to practice embodiments ofthe present invention is described with reference to FIGS. 1 and 2 . Asshown in FIG. 1 , the laser surgery system 10 may include a lasersource/assembly 12, a confocal detection assembly 14, a free-floatingmechanism 16, a scanning assembly 18, an objective lens assembly 20, anda patient interface device 22. The patient interface device 22 may beconfigured to interface with a patient's eye 24. The patient interfacedevice 22 may be supported by the objective lens assembly 20, which maybe supported by the scanning assembly 18, which may be supported by thefree-floating mechanism 16. The free-floating mechanism 16 may have aportion having a fixed position and orientation relative to the laserassembly 12 and the confocal detection assembly 14. The laser beam 28may propagate through the free-floating mechanism 16 along a variableoptical path 30, which may deliver the beam 28 to the scanning assembly18. An optical delivery system for receiving and directing the treatmentbeam may comprise some or all of the components coupled to the to thesub-nanosecond laser assembly 12. In some embodiments, the patientinterface device 22 can be configured to be coupled to an eye of the eye24 using vacuum. The laser surgery system 10 can further optionallyinclude a base assembly 26 that can be fixed in place or berepositionable.

The electromagnetic radiation beam 28 emitted by the laser assembly 12can include a series of laser pulses of any suitable energy level,duration, and repetition rate. In many embodiments, the laser assembly12 incorporates sub-nanosecond laser technology where a short duration(e.g., approximately 10 ns to 1 picosecond in duration) laser pulse(with energy level in the tens of micro joules range) can be deliveredto a tightly focused point to disrupt tissue, thereby substantiallylowering the energy level required to image and/or modify an intraoculartarget as compared to laser pulses having longer durations. The laserassembly 12 may produce laser pulses having a wavelength suitable totreat and/or image tissue.

The laser assembly 12 may include control and conditioning components.In an embodiment, the control components may include a beam attenuatorto control the energy of the laser pulse and the average power of thepulse train, a fixed aperture to control the cross-sectional spatialextent of the beam containing the laser pulses, one or more powermonitors to monitor the flux and repetition rate of the beam train andtherefore the energy of the laser pulses, and a shutter to allow/blocktransmission of the laser pulses. The conditioning components mayinclude an adjustable zoom assembly and a fixed optical relay totransfer the laser pulses over a distance while accommodating laserpulse beam positional and/or directional variability, thereby providingincreased tolerance for component variation.

In some embodiments, the scanning assembly 18 can include a Z-scandevice and an XY-scan device. The laser surgery system 10 may beconfigured to focus the electromagnetic radiation beam 28 to a focalpoint that is scanned in three dimensions. The Z-scan device may beoperable to vary the location of the focal point in the direction ofpropagation of the beam 28. The XY-scan device may be operable to scanthe location of the focal point in two dimensions transverse to thedirection of propagation of the beam 28. Accordingly, the combination ofthe Z-scan device and the XY-scan device can be operated to controllablyscan the focal point of the beam in three dimensions, including: withina tissue, e.g., eye tissue, of the eye 24. The scanning assembly 18 maybe supported by the free-floating mechanism 16, which may accommodatepatient movement, induced movement of the scanning assembly 18 relativeto the laser assembly 12 and the confocal detection assembly 14 in threedimensions.

FIG. 2 schematically illustrates details of an embodiment of the lasersurgery system 10. Specifically, example configurations areschematically illustrated for the laser assembly 12, the confocaldetection assembly 14, and the scanning assembly 18. As shown in theillustrated embodiment, the laser assembly 12 may include an IR laser32, alignment mirrors 34, 36, a beam expander 38, a one-half wave plate40, a polarizer and beam dump device 42, output pickoffs and monitors44, and a system-controlled shutter 46. The electromagnetic radiationbeam 28 output by the laser 32 may be deflected by the alignment mirrors34, 36. In many embodiments, the alignment mirrors 34, 36 may beadjustable in position and/or orientation so as to provide the abilityto align the beam 28 with the downstream optical path through thedownstream optical components. Next, the beam 28 may pass through thebeam expander 38, which can increase the diameter of the beam 28. Theexpanded beam 28 may then pass through the one-half wave plate 40 beforepassing through the polarizer 42. The beam exiting the polarizer 42 maybe linearly polarized. The one-half wave plate 40 can rotate thispolarization. The amount of light passing through the polarizer 42depends on the angle of the rotation of the linear polarization.Therefore, the one-half wave plate 40 with the polarizer 42 may act asan attenuator of the beam 28. The light rejected from this attenuationmay be directed into the beam dump. Next, the attenuated beam 28 maypass through the output pickoffs and monitors 44 and then through thesystem-controlled shutter 46. By locating the system-controlled shutter46 downstream of the output pickoffs and monitors 44, the power of thebeam 28 can be checked before opening the system-controlled shutter 46.

The system 10 can be set to locate the anterior and posterior surfacesof the lens capsule and cornea and ensure that the laser pulse beam 28will be focused on the lens capsule and cornea at all points of thedesired opening. In the embodiment of FIGS. 1 and 2 , a confocaldetection assembly 14 is described, although other modalities are withinthe scope of the present invention. Imaging systems and techniquesdescribed herein, such as for example, Optical Coherence Tomography(OCT), Purkinje imaging, Scheimpflug imaging, structured lightillumination, confocal backreflectance imaging, fluorescence imaging,ultrasound, or other ophthalmic or medical imaging modalities and/orcombinations thereof, may be used to measure structures of theanatomical components of the eye, such as to determine the location andmeasure the thickness of the lens and lens capsule to provide greaterprecision to the laser focusing methods. The imaging modalities canperform 2D and 3D patterning. For example, an OCT scan of the eye canprovide information about the shape of the cornea, the axial location ofthe anterior and posterior lens capsule, the boundaries of the cataractnucleus, as well as the depth of the anterior chamber. This informationis then loaded into the control electronics 70, and used to program andcontrol the subsequent laser-assisted surgical procedure. Theinformation may also be used to determine a wide variety of parametersrelated to the procedure such as, for example, the upper and lower axiallimits of the focal planes used for modifying the lens capsule, cornea,and synthetic intraocular lens implant, among others.

As shown in the illustrated embodiment, the scanning assembly 18 mayinclude a Z-scan device 58 and an XY-scan device 60. The Z-scan device58 may be operable to vary a convergence/divergence angle of the beam 28and thereby change a location of the focal point in the direction ofpropagation of the beam 28. For example, the Z-scan device 58 mayinclude one or more lenses that are controllably movable in thedirection of propagation of the beam 28 to vary a convergence/divergenceangle of the beam 28. The XY-scan device 60 may be operable to deflectthe beam 28 in two dimensions transverse to the direction of propagationof the beam 28. For example, the XY-scan device 60 can include one ormore mirrors that are controllably deflectable to scan the beam 28 intwo dimensions transverse to the direction of propagation of the beam28. Accordingly, the combination of the z-scan device 58 and the xy-scandevice 60 can be operated to controllably scan the focal point in threedimensions, for example, within the eye of the patient.

As shown further in the illustrated embodiment, a camera 62 andassociated video illumination 64 can be integrated with the scanningassembly 18. The camera 62 and the beam 28 may share a common opticalpath through the objective lens assembly 20 to the eye. A video dichroic66 may be used to combine/separate the beam 28 with/from theillumination wavelengths used by the camera. For example, the beam 28can have a wavelength of about 355 nm and the video illumination 64 canbe configured to emit illumination having wavelengths greater than 450nm. Accordingly, the video dichroic 66 can be configured to reflect the355 nm wavelength while transmitting wavelengths greater than 450 nm.

The control electronics 70 controls the operation of and can receiveinput from the laser assembly 12, the confocal detection assembly 14,free-floating mechanism 16, the scanning assembly 18, the objective lensassembly 20, the patient interface 22, control panel/graphical userinterface (GUI) 72, and user interface devices 74 via communicationpaths. The communication paths can be implemented in any suitableconfiguration, including any suitable shared or dedicated communicationpaths between the control electronics 70 and the respective systemcomponents.

The control electronics 70 can include any suitable components, such asone or more processors, one or more field-programmable gate array(FPGA), and one or more memory storage devices. The control electronics70 is operatively coupled via the communication paths with the laserassembly 12, the confocal detection assembly 14, the free-floatingmechanism 16, the scanning assembly 18, the control panel/GUI 72, andthe user interface devices 74. In many embodiments, the controlelectronics 70 controls the control panel/GUI 72 to provide forpre-procedure planning according to user specified treatment parametersas well as to provide user control over the laser eye surgery procedure.The control electronics 70 can include a processor/controller that isused to perform calculations related to system operation and providecontrol signals to the various system elements. A computer readablemedium can be coupled to the processor in order to store data used bythe processor and other system elements. The processor interacts withthe other components of the system as described more fully throughoutthe present specification. In an embodiment, the memory can include alook up table that can be utilized to control one or more components ofthe laser system surgery system.

The processor can be a general purpose microprocessor configured toexecute instructions and data such as a processor manufactured by theIntel Corporation of Santa Clara, Calif. It can also be an ApplicationSpecific Integrated Circuit (ASIC) that embodies at least part of theinstructions for performing the method according to the embodiments ofthe present disclosure in software, firmware and/or hardware. As anexample, such processors include dedicated circuitry, ASICs,combinatorial logic, other programmable processors, combinationsthereof, and the like. The memory can be local or distributed asappropriate to the particular application. Memory can include a numberof memories including a main random access memory (RAM) for storage ofinstructions and data during program execution and a read only memory(ROM) in which fixed instructions are stored. Thus, the memory providespersistent (non-volatile) storage for program and data files, and mayinclude a hard disk drive, flash memory, a floppy disk drive along withassociated removable media, a Compact Disk Read Only Memory (CD-ROM)drive, an optical drive, removable media cartridges, and other likestorage media.

In a laser cataract procedure according to embodiments of the presentinvention, an iris registration step is performed based on iris imagestaken before and after the eye is docked to the patient interface (PI)device, as summarized in the flow chart of FIG. 3 . First, an iris imageis taken while the eye is not docked to the PI (referred to as the“undocked” iris image) (step S11). This may be done, for example usingthe video camera of the laser system 10. The astigmatism is measured inthe undocked condition (step S12), for example, by using the OCT systemor other imaging system of the laser system 10. Alternatively, theastigmatism may be measured using a diagnostic system separate from thecataract laser system, in which case the undocked iris image may betaken by the separate diagnostic system as well, and the undocked irisimage and the measured astigmatism information are stored in a memory ofthe control computer of the cataract laser system or another computersystem and can be read out during the cataract procedure.

The eye is subsequently docked to the cataract laser system via the PIdevice (step S13). Another iris image is taken while the eye is dockedto the PI (referred to as the “docked” iris image) (step S14). Thecontrol computer of the cataract laser system performs an irisregistration process (step S15), by first comparing the undocked irisimage and the docked iris image to compute a transformation that mapspositions in the undocked iris image to positions in the docked irisimage, and then applying the transformation to the astigmatism axis ofthe undocked eye (measured in step S12) to transform it to a rotatedorientation relative to the docked iris image, which gives theastigmatism axis of the docked eye. The cataract procedure thenproceeds, including the astigmatism correction steps which is performedbased on the rotated astigmatism orientation relative to the docket irisimage (step S16). The astigmatism correction steps may include formingone or more arcuate incisions on the cornea or sclera, or implanting anIOL having astigmatism correction power; the positions of the arcuateincisions or the orientation of the IOL are dependent on the rotatedastigmatism axis orientation.

The iris registration step S15 finds the relative rotation of the eyebetween the undocked iris image and the docked iris image. To accomplishthis, a parameterized transformation is defined, which map pixelpositions in the undocked iris image to corresponding mapped pixelpositions in the docked iris image; the transformation is optimized suchthat an error term, which represents differences between pixel values ofthe undocked iris image at its pixel positions and pixel values of thedocked iris image at the corresponding mapped pixel positions, isminimized. This optimization can be thought of as transforming theundocked iris image into a transformed image that overlaps best with thedocked iris image. The iris registration algorithm according to anembodiment of the present invention is described with reference to FIG.4 .

First (step S21), a transformation is defined, which includes: atranslation mapping, which maps the pupil center in the undocked irisimage to the pupil center in the docked iris image; a dilation mapping,which accounts for pupil dilation (difference in pupil diameters in theundocked and docked iris images) and adjusts the pixel distance from thepupil center accordingly; and a rotation mapping, which represents arotation between the two iris images. The parameters of thetransformation that maximize the match between the two iris images arefound numerically. The rotation parameter represents the iris rotationangle which can then be used to determine the orientation of astigmatismaxis relative to the docked iris image.

More specifically, let {right arrow over (x_(u))} be a pixel in theundocked iris image, and let its gray level value be G_(u)({right arrowover (x_(u))})=G_(u)(x_(u), y_(u)); let {right arrow over (x_(d))} be aposition in the docked iris image that correspond to the pixel {rightarrow over (x_(u))} in the undocked iris image by the transformation{right arrow over (T)}, i.e., {right arrow over (x_(d))}={right arrowover (T)}({right arrow over (x_(u))}), and let the interpolated pixelgray level value at this position of the docked iris image beG_(d)({right arrow over (x_(d))})=G_(d)(x_(d), y_(d)). The gray levelvalue G_(d)({right arrow over (x_(d))}) in the docked iris image iscompared with the gray level value G_(u)({right arrow over (x_(u))}) ofthe undocked iris image to find the transformation parameters thatmaximize the match between them.

In a preferred embodiment, the transformation {right arrow over (T)}that maps a pixel position in the undocked iris image to a position inthe docked iris image is defined as (Eq. (1)):

$\overset{\rightarrow}{x_{d}} = {\begin{Bmatrix}x_{d} \\y_{d}\end{Bmatrix} = {\begin{Bmatrix}{T_{x}\left( \overset{\rightarrow}{x_{u}} \right)} \\{T_{y}\left( \overset{\rightarrow}{x_{u}} \right)}\end{Bmatrix} = {{\overset{\rightarrow}{T}\left( \overset{\rightarrow}{x_{u}} \right)} = {{\overset{\rightarrow}{T}\left( \begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} \right)} = {{\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)*\begin{bmatrix}{\cos(\theta)} & {\sin(\theta)} \\{{- \sin}(\theta)} & {\cos(\theta)}\end{bmatrix}\ \left\{ {\begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} - \ \begin{Bmatrix}x_{{center} - u} \\y_{{center} - u}\end{Bmatrix}} \right\}} + \begin{Bmatrix}x_{{center}‐d} \\y_{{center}‐d}\end{Bmatrix}}}}}}$

where e is a pupil dilation parameter, θ is an iris rotation parameter,x_(center-d) and y_(center-d) are the x and y coordinates of the pupilcenter in the docked iris image, respectively, x_(center-u) andy_(center-u) are the x and y coordinates of the pupil center in theundocked iris image, respectively, L is the limbus radius, and R is theradial position of the pixel {right arrow over (x_(u))} in the undockediris image with respect to the pupil center, i.e.

R=√{square root over ((x _(u) −x _(center-u))²+(y _(u) −y_(center-u))²)}

The transformation is optimized for the parameters e (dilation), θ(rotation), and x_(center-d) and y_(center-d) (translation), while theparameters x_(center-u), y_(center-u), and L stay constant during theoptimization process.

In the transformation T defined in Eq. (1), the factor

$\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)$

compensates for the effect of pupil dilation (the term dilation is usedhere to include both dilation and constriction of the pupil); it maps adistance R of a pixel from the iris center to a distance

$\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right).$

At R=L (i.e. at the radius of the limbus), the mapped distance isunchanged regardless of the value e. In other words, the dilationmapping preserves the limbus location. At R<L (i.e. inside the limbus),the mapped distance is greater than R for e>0 (pupil is dilated) or lessthan R for e<0 (pupil is constricted). When e=0, mapped distance isunchanged for all distance R. It should be noted that this dilationfactor is not an overall scaling of the iris image; rather, itrepresents the fact that when pupil size changes, the limbus locationdoes not change but the positions of the iris patterns, which arelocated between the pupil and the limbus, change.

The x and y coordinates of the pupil center of the undocked iris image,x_(center-u), y_(center-u), are computed by fitting the pupil boundaryin the undocked iris image to a circle or an ellipse. The limbus radiusis computed by fitting the limbus boundary to a circle or an ellipse. Ifit is fitted to an ellipse, the limbus radius will have two values, onedefined along the major axis and one defined along the minor axis. Insuch a case, the pupil dilation factor can be modified accordingly. Anysuitable method may be used to fit the limbus and pupil boundaries. Forexample, an edge detection algorithm may be used to identify the pupiland limbus boundaries in the images.

FIGS. 5A-5H are examples that illustrate the effect of the variousfactors in the transformation. The left ones of the image pairs, namelyFIGS. 5A, 5C, 5E, and 5G, are the undocked iris images (they areidentical), and the right ones of the image pairs, namely FIGS. 5B, 5D,5F, and 5H, are docked iris images. In FIGS. 5A and 5B, the dark circlespointed to by the thick arrows are the pupil boundaries. An oval shape11 is overlaid in the undocked image FIG. 5A that fits the pupilboundary. The oval shape 12 overlaid in the docked image is at the samelocation as the oval 11 in the undocked image, but it does not coincidewith the pupil boundary as the two images are not registered with eachother. FIG. 5D shows the effect of translation, where the docked imageis translated to the right as indicated by the thick arrows, and thepupil boundary is now more centered with respect to the oval 12. FIG. 5Fshows the effect of pupil dilation, where the docked image is appliedwith the dilation factor as indicated by the thick arrows, and theresulting pupil boundary is now approximately the same size as the oval12. FIG. 5H shows the effect of rotation, where the docked image isrotated clockwise as indicated by the thick arrows. It should be notedthat FIGS. 5A-5H are only intended to aid in understanding of the effectof the transformation factors; in the actual data processing, describedin more detail below, the translation, dilation and rotation operationsare typically not applied in separate steps in the manner illustrated inFIGS. 5A-5H.

Next (step S22), the transformation is optimized by minimizing an errorterm. In a preferred embodiment, the optimization of the transformationparameters is done using the Newton-Raphson method to find values of theparameters (θ, e, x_(center-d), y_(center-d)) that minimize the squareerror term ϵ (Eq. (2)):

${\epsilon\left( {\theta,e,x_{{center} - d},y_{{center} - d}} \right)} = {{\sum\limits_{i = 1}^{pixels}\left( {{G_{u}\left( \overset{\rightarrow}{x_{u_{i}}} \right)} - {G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}} \right)^{2}} = {\sum\limits_{i = 1}^{pixels}\left( {{G_{u}\left( \overset{\rightarrow}{x_{u_{i}}} \right)} - {G_{d}\left( {\overset{\rightarrow}{T}\left( \overset{\rightarrow}{x_{u_{i}}} \right)} \right)}} \right)^{2}}}$

The summation here is over all pixels in the undocked iris image withina defined area of interest. In preferred embodiments, the area ofinterest is the area located between the limbus and a circle interior tothe pupil.

The minimization starts with a set of initial parameter values, whichmay be zero or non-zero. In each iteration of the Newton-Raphsoniterative method, first take the gradient of the error term ∈ withrespect to each of the four parameters P_(j), j=1, . . . 4 (P₁=θ, P₂=e,P₃=x_(center-d), P₄=y_(center-d)) at the current parameter values, andexpress the gradient in a component notation (Eq. (3)):

$\left( {\nabla\epsilon_{\overset{\rightarrow}{p}}} \right)_{j} = {{- 2}{\underset{i = 1}{\sum\limits^{pixels}}\left\lbrack {\left( {{G_{u}\left( \overset{\rightarrow}{x_{u_{i}}} \right)} - {G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}} \right)\left( {{\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial x}\frac{\partial{T_{x}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}}} + {\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial y}\frac{\partial{T_{y}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}}}} \right)} \right\rbrack}}$

Then take the gradient again with respect to each the four parametersP_(k), k=1, . . . 4 at the current parameter values to construct theHessian matrix (Eq. (4)):

$\left( {\nabla{\nabla\epsilon_{\overset{\rightarrow}{p}}}} \right)_{jk} = {{- 2}{\underset{i = 1}{\sum\limits^{pixels}}\left\lbrack {\left( {{\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial x}\frac{\partial{T_{x}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{k}}} + {\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial y}\frac{\partial{T_{y}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{k}}}} \right)\left( {{\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial x}\frac{\partial{T_{x}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}}} + {\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial y}\frac{\partial{T_{y}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}}}} \right)} \right\rbrack}}$

According to the Newton-Raphson iterative process, the gradient of theerror term, ∇∈_(p) _(j) , is expanded in a Taylor series and set equalto zero for a minimum error term (Eq. (5)):

0 = (∇ϵ_(p))_(j) = (∇ϵ_(p)❘_(P = P0))_(j) + (∇∇ϵ_(p)❘_(P = P0))_(jk){P_(0k) − P_(k)}

which lead to the equation that gives the updated parameters for aniteration (Eq. (5)):

P_(k) = P_(0k) + (∇∇ϵ_(p)❘_(P = P0))_(jk)⁻¹(∇ϵ_(p)❘_(P = P0))_(j)

where P_(0k) are the current parameter values for the current iterationand P_(k) are the updated parameter values resulting from the currentiteration, which will be used in the next iteration. The iterations arerepeated until the parameters converge to a set of values that minimizesthe square error term ϵ.

Based on the above principle, the minimization process using thegradient Newton-Raphson method include the following steps at eachiteration n. First, for every pixel i in the undocked iris image,located at pixel position {right arrow over (x_(u) _(i) )}, thecorresponding position {right arrow over (x_(d) _(i) )} in the dockediris image is computed using the transformation {right arrow over (T)}with the current parameters P_(n).

Then, the pixel intensity for pixel i in the undocked iris image,G_(u)({right arrow over (x_(u) _(i) )}) is obtained; the pixel intensityat the corresponding position in the docked iris image, G_(d)({rightarrow over (x_(d) _(i) )}) is computed using an interpolation method.Any suitable interpolation algorithm may be used; one example is abilinear interpolation using the four nearest pixels. A comparisonvector, i.e. the difference between the two pixel intensities,b_(i)=G_(d)({right arrow over (x_(d) _(i) )})−G_(u)({right arrow over(x_(u) _(i) )}), is constructed.

Also, for each pixel i, the first derivatives with respect to x and y ofthe pixel intensity of the docked iris image at the correspondingposition

$\overset{\rightarrow}{x_{d_{i}}},{i.e.},{\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial x}{and}\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial y}},$

are computed, using the interpolation in the previous step.

For each pixel i, the gradients of the transformation {right arrow over(T)} in the x and y directions with respect to the parameters

P_(j) = (θ, e, x_(center − d), y_(center − d)),${i.e.},{\left( \frac{\partial{T_{x}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}} \right)_{i}{and}\left( \frac{\partial{T_{y}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}} \right)_{i}},{j = 1},$

are also computed.

Then, the Hessian matrix is computed using the first derivatives andgradients computed above, i.e.,

$H_{ij} = {{\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial x}\frac{\partial{T_{x}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}}} + {\frac{\partial{G_{d}\left( \overset{\rightarrow}{x_{d_{i}}} \right)}}{\partial y}\frac{\partial{T_{y}\left( \overset{\rightarrow}{x_{u}} \right)}}{\partial P_{j}}}}$

Next, the Newton-Raphson system is constructed and solved (where j, k=1,. . . 4 are the indices of the parameters):

(I*λ+H _(ij) H _(ik))ΔP _(j) =H _(ik) b _(i)

In the above Newton-Raphson system, to improve convergence, a dampingparameter (Levenberg-Marquardt parameter) λ is used. I is the identitymatrix.

Finally, the transformation parameters are updated:

P _(n+1) =P _(n) +ΔP

The above steps are repeated for each iteration until the parametervalues converge. A predefined threshold may be used to determineconvergence.

Once the transformation is optimized, the rotation angle θ of thetransformation is obtained, and the astigmatism axis measured before eyedocking is rotated by that rotation angle to obtain the rotatedastigmatism axis relative to the docked iris image (step S23).

The convergence radius of the above minimization method can be small,and the process may converge to a local minimum easily but the localminimum may not necessarily be the correct transformation. To overcomethis difficulty, in some embodiments, the minimization process isperformed several times, each time with a different initial rotationparameter (and optionally, different initial translation and dilationparameters). When all minimizations are completed, if several of themconverge to the same rotation angle, with a small error term, thisrotation angle will be deemed the correct rotation angle. Thisredundancy provides assurance that the result is correct.

The minimization process is less sensitive to local minimums whencoarsed (lower spatial resolution) images are used. A coarsed image maybe generated, for example, by averaging every N by N pixel area of theoriginal (undocked or docked) iris images into a single pixel. Theminimization process also runs faster on coarsed images. Therefore, insome embodiments, a coarsed undocked iris image and a coarsed dockediris image may be used in an initial minimization process to findinitial conditions, which are then used to perform the minimizationprocess using higher resolution (e.g. the original full resolution)images. In some embodiments, a pyramid method may be used where theminimization process is repeated on progressively higher resolutionimages (all generated from the original full resolution images), eachtime using the parameters found in the previous repetition as theinitial conditions. Each of the repetitions uses the same minimizationalgorithm described above but with different images and differentinitial parameters.

When applying the minimization method to iris images, care should betaken to avoid certain artifacts peculiar to iris images. For example,in a typical undocked iris image, portions of the iris may be covered bythe eyelids, whereas in the docked iris image, these portions arevisible because the eyelids are kept away by the PI device. Therefore,the two iris images will not match each other in these portions. Toaddress this problem, a preferred embodiment of the present inventionincludes an outlier adjustment or rejection step. More specifically, ineach iteration of the Newton-Raphson minimization process describedabove, for each pixel in the undocked iris image, the pixel intensitydifference between that pixel and the corresponding pixel at the mappedposition in the docked iris image is evaluated; pixel pairs that havecomparatively greater intensity differences are deemed outliers andassigned comparatively lower weights in the error term c in theiteration. Any appropriate criteria may be used to determine outliersand to adjust the weights of the pixels. In one embodiment, pixels thathave intensity differences greater than a predefined threshold (e.g., apredefined multiplier of the mean intensity difference of all pixels)are assigned a weight of zero (i.e. they are excluded). This methodworks well to identify eyelids and reduce their deleterious effect onprecision and convergence.

As described earlier, in some embodiments, in the undocked iris images,only pixels located between the limbus and the iris/pupil boundary areused in the optimization process. In some embodiments, some pixelsinside the pupil are included in the optimization process, which allowsfor comparison of pupil shapes as well.

It should be noted that since the docked and undocked iris images aresymmetrical in their roles, the transformation may be applied to mappositions in the docked iris image to positions in the undocked irisimage, i.e., {right arrow over (x_(u))}={right arrow over (T)}({rightarrow over (x_(d))}), which gives an equivalent result with thecalculated rotation angle being a rotation in the opposite direction.

The iris registration method described above may be applied in otherlaser ophthalmic surgery procedures for astigmatism correction where apatient interface is used to dock the eye, such as small incision lensextraction.

It will be apparent to those skilled in the art that variousmodification and variations can be made in the iris registration methodand related apparatus of the present invention without departing fromthe spirit or scope of the invention. Thus, it is intended that thepresent invention cover modifications and variations that come withinthe scope of the appended claims and their equivalents.

1. A laser ophthalmic surgery system for treating a patient's eye,comprising: a laser source configured to generate a pulsed laser beam;an optical delivery system coupled to the laser source, and configuredto receive and direct the pulsed laser beam; a camera coupled to theoptical delivery system and configured to obtain images of the eye; anda processor coupled to the laser source, the optical delivery system,and the camera, the processor comprising a non-transitory computerreadable medium storing computer executable instructions configured toinstruct the processor to perform a process which includes: obtaining anundocked iris image of the eye, the undocked iris image having beentaken when the eye is not mechanically coupled to any patient interfacedevice; obtaining a measured geometric characteristic of the eye, themeasured geometric characteristic having been measured when the eye isnot mechanically coupled to any patient interface device; controllingthe camera to take a docked iris image of the eye when the eye ismechanically coupled to a patient interface device; defining aparameterized transformation which maps pixel positions in the undockediris image to corresponding mapped pixel positions in the docked irisimage, wherein the parameterized transformation includes a translationmapping using translation parameters, a dilation mapping using a pupildilation parameter, and a rotation mapping using a rotation angleparameter, wherein the dilation mapping maps a distance between a pixeland a pupil center to a mapped distance based on the dilation parameterwhile mapping a limbus radius of a limbus of the eye to the limbusradius itself; and optimizing the transformation by minimizing an errorterm that represents a difference between pixel values of the undockediris image at its pixel positions and pixel values of the docked irisimage at the corresponding mapped pixel positions, to obtain anoptimized transformation; computing a transformed geometriccharacteristic of the eye by applying the optimized transformation tothe measured geometric characteristic; and while the eye is mechanicallycoupled to the patient interface device, controlling the laser sourceand the optical delivery system based on the transformed geometriccharacteristic of the eye to deliver the pulsed laser beam into the eyeto treat the eye.
 2. The laser ophthalmic surgery system of claim 1,wherein the transformation is defined as: $\begin{Bmatrix}x_{d} \\y_{d}\end{Bmatrix} = {{\overset{\rightarrow}{T}\left( \begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} \right)} = {{\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)*\begin{bmatrix}{\cos(\theta)} & {\sin(\theta)} \\{{- \sin}(\theta)} & {\cos(\theta)}\end{bmatrix}\ \left\{ {\begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} - \ \begin{Bmatrix}x_{{center} - u} \\y_{{center} - u}\end{Bmatrix}} \right\}} + \begin{Bmatrix}x_{{center} - d} \\y_{{center} - d}\end{Bmatrix}}}$ where x_(u) and y_(u) are x and y coordinates of apixel position in the undocked iris image, x_(d) and y_(d) are x and ycoordinates of a corresponding transformed pixel position, e is thepupil dilation parameter and$\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)$ is thedilation mapping, θ is the rotation angle parameter, x_(center-u) andy_(center-u) are the translation parameters representing x and ycoordinates of a pupil center in the undocked iris image, x_(center-d)and y_(center-d) are x and y coordinates of the pupil center in thedocked iris image, L is the limbus radius, andR=√{square root over ((x _(u) −x _(center-u))²+(y _(u) −y_(center-u))²)} is a radial position of the pixel in the undocked irisimage with respect to the pupil center.
 3. The laser ophthalmic surgerysystem of claim 1, wherein the step of optimizing the transformation isperformed using a set of initial parameter values and a gradientNewton-Raphson iterative method.
 4. The laser ophthalmic surgery systemof claim 3, wherein the gradient Newton-Raphson iterative methodincludes a damping parameter.
 5. The laser ophthalmic surgery system ofclaim 1, wherein the step of optimizing the transformation includes:providing a plurality of different sets of initial parameter values;performing a gradient Newton-Raphson iterative method a plurality oftimes, each time using one of the plurality of different sets of initialparameter values, to obtain a plurality of sets of candidate optimizedvalues of the transformation parameters; and determining optimizedvalues of the transformation parameters based on the plurality of setsof candidate optimized values.
 6. The laser ophthalmic surgery system ofclaim 1, wherein the step of optimizing the transformation includes:generating a first reduced-resolution iris image from the undocked irisimage, and generating a second reduced-resolution iris image from thedocked iris image, the first and second reduced-resolution iris imageshaving lower resolution than the undocked and docked iris images,respectively; optimizing the transformation by minimizing an error termthat represents a difference between pixel values of the firstreduced-resolution iris image at its pixel positions and pixel values ofthe second reduced-resolution iris image at the corresponding mappedpixel positions, to obtain a first set of optimized values of thetransformation parameters; and using the first set of optimized valuesof the transformation parameters as initial parameter values, furtheroptimizing the transformation by minimizing an error term thatrepresents a difference between pixel values of the undocked iris imageat its pixel positions and pixel values of the docked iris image at thecorresponding mapped pixel positions, to obtain a second set ofoptimized values of the transformation parameters.
 7. The laserophthalmic surgery system of claim 1, wherein the error term iscalculated as a sum, over pixels of the undocked iris image, of aweighted square of a pixel value difference between a pixel value of theundocked iris image at its pixel position and a pixel value of thedocked iris image at the corresponding mapped pixel position, wherein aweight assigned to each square of pixel value difference in the sum isdetermined based on the pixel value difference, wherein the weight islower for higher pixel value difference, and wherein the weight is zerowhen the pixel value difference exceeds a predefined threshold.
 8. Thelaser ophthalmic surgery system of claim 1, wherein the error term iscalculated as a sum, over pixels within a predefined area of theundocked iris image, of a weighted square of a pixel value differencebetween a pixel value of the undocked iris image at its pixel positionand a pixel value of the docked iris image at the corresponding mappedpixel position, wherein the predefined area is located between thelimbus of the eye and a pupil boundary of the eye, or located betweenthe limbus and a circle within the pupil boundary.
 9. The laserophthalmic surgery system of claim 1, further comprising an imagingsystem configured to measure structures of anatomical components of theeye, wherein the step of obtaining the undocked iris image of the eyeincludes the processor controlling the camera to obtain the undockediris image, and the step of obtaining the measured geometriccharacteristic of the eye includes the processor controlling the imagingsystem to measure the geometric characteristic.
 10. The laser ophthalmicsurgery system of claim 1, wherein the step of obtaining the undockediris image of the eye includes the processor reading the undocked irisimage from a memory of the processor, and the step of obtaining themeasured geometric characteristic of the eye includes the processorreading the measured geometric characteristic from a memory of theprocessor.
 11. A method for treating a patient's eye, implemented in alaser ophthalmic surgery system which includes a processor, the methodcomprising a plurality of steps performed by the processor including:obtaining an undocked iris image of the eye, the undocked iris imagehaving been taken when the eye is not mechanically coupled to anypatient interface device; obtaining a measured geometric characteristicof the eye, the measured geometric characteristic having been measuredwhen the eye is not mechanically coupled to any patient interfacedevice; controlling a camera of the laser ophthalmic surgery system totake a docked iris image of the eye when the eye is mechanically coupledto a patient interface device; defining a parameterized transformationwhich maps pixel positions in the undocked iris image to correspondingmapped pixel positions in the docked iris image, wherein theparameterized transformation includes a translation mapping usingtranslation parameters, a dilation mapping using a pupil dilationparameter, and a rotation mapping using a rotation angle parameter,wherein the dilation mapping maps a distance between a pixel and a pupilcenter to a mapped distance based on the dilation parameter whilemapping a limbus radius of a limbus of the eye to the limbus radiusitself; and optimizing the transformation by minimizing an error termthat represents a difference between pixel values of the undocked irisimage at its pixel positions and pixel values of the docked iris imageat the corresponding mapped pixel positions, to obtain an optimizedtransformation; computing a transformed geometric characteristic of theeye by applying the optimized transformation to the measured geometriccharacteristic; and while the eye is mechanically coupled to the patientinterface device, controlling a laser source and an optical deliverysystem of the laser ophthalmic surgery system based on the transformedgeometric characteristic of the eye to deliver a pulsed laser beam intothe eye to treat the eye.
 12. The method of claim 11, wherein thetransformation is defined as: $\begin{Bmatrix}x_{d} \\y_{d}\end{Bmatrix} = {{\overset{\rightarrow}{T}\left( \begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} \right)} = {{\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)*\begin{bmatrix}{\cos(\theta)} & {\sin(\theta)} \\{{- \sin}(\theta)} & {\cos(\theta)}\end{bmatrix}\ \left\{ {\begin{Bmatrix}x_{u} \\y_{u}\end{Bmatrix} - \ \begin{Bmatrix}x_{{center} - u} \\y_{{center} - u}\end{Bmatrix}} \right\}} + \begin{Bmatrix}x_{{center} - d} \\y_{{center} - d}\end{Bmatrix}}}$ where x_(u) and y are x and y coordinates of a pixelposition in the undocked iris image, x_(d) and y_(d) are x and ycoordinates of a corresponding transformed pixel position, e is thepupil dilation parameter and$\left( {1 + {e\left( {\frac{L}{R} - 1} \right)}} \right)$ is thedilation mapping, θ is the rotation angle parameter, x_(center-u) andy_(center-u) are the translation parameters representing x and ycoordinates of a pupil center in the undocked iris image, x_(center-d)and y_(center-d) are x and y coordinates of the pupil center in thedocked iris image, L is the limbus radius, andR=√{square root over ((x _(u) −x _(center-u))²+(y _(u) −y_(center-u))²)} is a radial position of the pixel in the undocked irisimage with respect to the pupil center.
 13. The method of claim 11,wherein the step of optimizing the transformation is performed using aset of initial parameter values and a gradient Newton-Raphson iterativemethod.
 14. The method of claim 13, wherein the gradient Newton-Raphsoniterative method includes a damping parameter.
 15. The method of claim11, wherein the step of optimizing the transformation includes:providing a plurality of different sets of initial parameter values;performing a gradient Newton-Raphson iterative method a plurality oftimes, each time using one of the plurality of different sets of initialparameter values, to obtain a plurality of sets of candidate optimizedvalues of the transformation parameters; and determining optimizedvalues of the transformation parameters based on the plurality of setsof candidate optimized values.
 16. The method of claim 11, wherein thestep of optimizing the transformation includes: generating a firstreduced-resolution iris image from the undocked iris image, andgenerating a second reduced-resolution iris image from the docked irisimage, the first and second reduced-resolution iris images having lowerresolution than the undocked and docked iris images, respectively;optimizing the transformation by minimizing an error term thatrepresents a difference between pixel values of the firstreduced-resolution iris image at its pixel positions and pixel values ofthe second reduced-resolution iris image at the corresponding mappedpixel positions, to obtain a first set of optimized values of thetransformation parameters; and using the first set of optimized valuesof the transformation parameters as initial parameter values, furtheroptimizing the transformation by minimizing an error term thatrepresents a difference between pixel values of the undocked iris imageat its pixel positions and pixel values of the docked iris image at thecorresponding mapped pixel positions, to obtain a second set ofoptimized values of the transformation parameters.
 17. The method ofclaim 11, wherein the error term is calculated as a sum, over pixels ofthe undocked iris image, of a weighted square of a pixel valuedifference between a pixel value of the undocked iris image at its pixelposition and a pixel value of the docked iris image at the correspondingmapped pixel position, wherein a weight assigned to each square of pixelvalue difference in the sum is determined based on the pixel valuedifference, wherein the weight is lower for higher pixel valuedifference, and wherein the weight is zero when the pixel valuedifference exceeds a predefined threshold.
 18. The method of claim 11,wherein the error term is calculated as a sum, over pixels within apredefined area of the undocked iris image, of a weighted square of apixel value difference between a pixel value of the undocked iris imageat its pixel position and a pixel value of the docked iris image at thecorresponding mapped pixel position, wherein the predefined area islocated between the limbus of the eye and a pupil boundary of the eye,or located between the limbus and a circle within the pupil boundary.19. The method of claim 11, wherein the step of obtaining the undockediris image of the eye includes using the camera to obtain the undockediris image, and the step of obtaining the measured geometriccharacteristic of the eye includes using an imaging system of the laserophthalmic surgery system to measure the geometric characteristic. 20.The method of claim 11, wherein the step of obtaining the undocked irisimage of the eye includes reading the undocked iris image from a memory,and the step of obtaining the measured geometric characteristic of theeye includes reading the measured geometric characteristic from amemory.